If $T \in L(V)$ is normal, does there exist an orthonormal basis $\beta$ of eigenvectors for T such that $[T]_\beta$ is diagonal?

Question

If $T \in L(V)$ is normal, does there exist an orthonormal basis $\beta$ of eigenvectors for T such that $[T]_\beta$ is diagonal?

If F=C, yes,

If F=R, no. $\rightarrow A= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \rightarrow $ when polynomial$ t^2+1 \rightarrow $no real eigenvalues $\rightarrow$ not diagonalizable over R.

I don't understand what does orthonormal basis play in here.